Mathematical Foundations of Computer Science

Prof. Dr. Erich Grädel

Algorithmic Model Theory

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Summary

Finite model theory studies the relationship between logical
definability and computational complexity on finite
structures. A particularly important aspect concerns logical
descriptions of complexity classes. Our research group has
made significant contributions to this area.

A newer development in this field is the extension of the
approach and methodology of finite model theory to (particular
classes of) infinite structures. Algorithmic issues on
infinite structures are of increasing importance in several
areas of computer science. In databases, the traditional
model based on finite relational structures has turned out to
be inadequate for modern applications (like geographic data,
constraint databases, data on the Web). Also in verification,
infinite (but finitely presentable) transition systems become
more and more important, in particular for applications to
software.

We investigate several directions, for making the methodology
developed in finite model theory applicable to infinite
structures. Of particular importance are, again, the
connections between algorithmic issues and logical
definability.

We have developed a model theory of metafinite
structures that combine finite structures with
arithmetic operations on infinite numerical domains.
Applications of metafinite model theory have been studied in
the following domains: descriptive complexity on real
numbers, approximation properties of optimization and
counting problems, databases with uncertain or unreliable
information, and database query languages with aggregates.

We study algorithmic and definability issues on various
classes of infinite structures that are presentable by
automata and interpretations. The
work by A. Blumensath, V. Bárány, and
E. Grädel on automatic structures has
been very influential for the development of this field.